Differential cross section for Compton scattering

Sarah Alanazi

Jacan Chaplais

April, 2021

Outline

  • Motivation
  • Theory
    • Frames of reference
    • The invariant amplitude
    • Phase space integrals
  • Method

Motivation

Quantum theory: old and new

In 1927, Dirac proposed that quantum mechanics may be elevated to a relativistic theory by quantizing a spinor field [1].

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Theory

Kinematics

During this calculation, we shall make use of two different inertial reference frames to obtain our final result. Mandelstam variables are constructed from the incoming and outgoing momenta of the interaction. \[ \begin{array}{lllllllll} s &=& (p+k)^{2} &=& p^{2}+k^{2}+2 p \cdot k &=& m^{2}+2 p \cdot k &=& m^{2}+2 p^{\prime} \cdot k^{\prime} \\ t &=& \left(p^{\prime}-p\right)^{2} &=& p^{\prime 2}+p^{2}-2 p \cdot p^{\prime} &=& 2 m^{2}-2 p \cdot p^{\prime} &=& -2 k \cdot k^{\prime} \\ u &=& \left(k^{\prime}-p\right)^{2} &=& k^{\prime 2}+p^{2}-2 k^{\prime} \cdot p &=& m^{2}-2 k^{\prime} \cdot p &=& m^{2}-2 k \cdot p^{\prime} \end{array} \] Casting transition amplitudes in terms of these manifestly Lorentz invariant quantities allows us to jump between frames with ease.

Centre-of-mass frame

Inetial frame in which sum of spatial momenta is zero.

Centre of mass frame diagram

Lab frame

Inertial frame in which the electron is at rest. This will also be the assumed rest frame for our particle detectors, hence lab frame.

Lab frame diagram

Differential cross section

blah blah

\[ \begin{aligned} \mathrm{d} \sigma=& \frac{1}{2 E_{\mathcal{A}} 2 E_{\mathcal{B}}\left|v_{\mathcal{A}}-v_{\mathcal{B}}\right|}\left(\prod_{f} \frac{\mathrm{d}^{3} p_{f}}{(2 \pi)^{3}} \frac{1}{2 E_{f}}\right) \\ &\left|\mathcal{M}\left(p_{\mathcal{A}}, p_{\mathcal{B}} \rightarrow\left\{p_{f}\right\}\right)\right|^{2}(2 \pi)^{4} \delta^{(4)}\left(p_{\mathcal{A}}+p_{\mathcal{B}}-\sum p_{f}\right) \end{aligned} \]

Phase space integral

\[ \int \mathrm{d} \Pi_{2}=\int \frac{\mathrm{d}^{3} k^{\prime}} {(2 \pi)^{3} 2 E_{k^{\prime}}} \frac{\mathrm{d}^{3} p^{\prime}}{(2 \pi)^{3} 2 E_{p^{\prime}}}(2 \pi)^{4} \delta^{4}\left(p+k-k^{\prime}-p^{\prime}\right) \]

Invariant amplitude

By applying the Feynman rules to these diagrams and grouping terms, we obtain \[ i \mathcal{M}=i e^{2} \epsilon_{\mu \lambda^\prime}^{\ast} \left(k^{\prime}\right) \epsilon_{\nu \lambda}\left(k\right) \bar{u}^{s^{\prime}}\left(p^{\prime}\right) \left( \dfrac{% numerator \gamma^{\mu}(\not{p}+k+m) \gamma^{\nu}} {(p+k)^{2}-m^{2}}% denominator + \dfrac{% numerator \gamma^{\nu} \left(\not{p}-k^{\prime}+m\right) \gamma^{\mu}} {\left(p-k^{\prime}\right)^{2}-m^{2}}% denominator \right) u^{s}(p) \]

This unwieldy expression can be simplified a little by expanding the binomials in the denominator, and observing for the numerator \[ \begin{aligned} (\not{p}+m) \gamma^{\nu} u^{s}(p) &=\left(2 p^{\nu}-\gamma^{\nu} \not{p}+\gamma^{\nu} m\right) u^{s}(p) \\ &=2 p^{\nu} u^{s}(p)-\gamma^{\nu}\underbrace{(\not{p}-m) u^{s}(p)}_{ \text{Dirac equation} \implies 0 } \\ &=2_{} p^{\nu} u^{s}(p) \end{aligned} \]

Evaluating invariant amplitude

Spin averages and polarisation sums

Blah blah algebra

Working out the traces

Bluh bluh traces

Phase space integral

In centre-of-mass frame

blah blah

In lab frame

dooda dooda

Obtaining the cross section

Function of squared momentum transfer

Function of angle

Results

QED prediction for \(\rm{d}\sigma/\rm{d}\theta\)

QED prediction for \(\rm{d}\sigma/\rm{d}t\)

Conclusion

References

[1] Dirac PAM. Quantum theory of emission and absorption of radiation. Proc Roy Soc Lond A 1927;114:243. https://doi.org/10.1098/rspa.1927.0039.